Fused Hecke algebra and one-boundary algebras

Loïc Poulain d’Andecy and Meri Zaimi

Vol. 328 (2024), No. 1, 77–118

DOI: 10.2140/pjm.2024.328.77

Abstract

This paper gives an algebraic presentation of the fused Hecke algebra which describes the centraliser of tensor products of the $U_q({\mathrm{gl}<!--FUNCTION\; APPLICATION-->}_N)$-representation labelled by a one-row partition of any size with vector representations. It is obtained through a detailed study of a new algebra that we call the symmetric one-boundary Hecke algebra. In particular, we prove that the symmetric one-boundary Hecke algebra is free over a ring of Laurent polynomials in three variables and we provide a basis indexed by a certain subset of signed permutations. We show how the symmetric one-boundary Hecke algebra admits the one-boundary Temperley–Lieb algebra as a quotient, and we also describe a basis of this latter algebra combinatorially in terms of signed permutations with avoiding patterns. The quotients corresponding to any value of $N$ in ${\mathrm{gl}<!--FUNCTION\; APPLICATION-->}_N$ (the Temperley–Lieb one corresponds to $N=2$) are also introduced. Finally, we obtain the fused Hecke algebra, and in turn the centralisers for any value of $N$, by specialising and quotienting the symmetric one-boundary Hecke algebra. In particular, this generalises to the Hecke case the description of the so-called boundary seam algebra, which is then obtained (taking $N=2$) as a quotient of the fused Hecke algebra.

Keywords

Hecke algebras, one-boundary, quantum groups, centraliser, fusion, signed permutations

Mathematical Subject Classification

Primary: 20C08

Secondary: 05E10, 20F36, 20G42

Milestones

Received: 7 August 2023

Revised: 15 December 2023

Accepted: 24 February 2024

Published: 16 April 2024